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With the advent of high-throughput technologies, ℓ 1 regularized learning algorithms have attracted much attention recently. Dozens of algorithms have been proposed for fast implementation, using various advanced optimization techniques. In this paper, we demonstrate that ℓ 1 regularized learning problems can be easily solved by using gradient-descent techniques. The basic idea is to transform a convex optimization problem with a non-differentiable objective function into an unconstrained non-convex problem, upon which, via gradient descent, reaching a globally optimum solution is guaranteed. We present detailed implementation of the algorithm using ℓ 1 regularized logistic regression as a particular application. We conduct large-scale experiments to compare the new approach with other stateof-the-art algorithms on eight medium and large-scale problems. We demonstrate that our algorithm, though simple, performs similarly or even better than other advanced algorithms in terms of computational efficiency and memory usage.

In this paper, the set-membership filtering problem is discussed for a class of discrete time-varying nonlinear complex networks with mixed time-delays and incomplete measurements. Both the process noise and the measurement noise are unknown but bounded. The nonlinear function is assumed to satisfy the sector bounded condition and the incomplete measurement is model by a diagonal matrix. The main purpose of this paper is to design a set-membership filtering method in order to provide the certain regions, which include the real states. Accordingly, the filtering errors at each time can be limited to the ellipsoid sets for the considered nonlinear delayed complex networks in the simultaneous presence of unknown but bounded noise, mixed time-delays and incomplete measurements. Moreover, a convex optimization method is presented to obtain the minimum ellipsoid domain. In addition, the filter gain matrix can be obtained by solving a series of linear matrix inequalities. Finally, a simulation example is used to illustrate the effectiveness of the proposed set-membership filtering method.

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